劉興文

(西南民族大學電氣信息工程學院，四川 成都 610041)

穩(wěn)定性是動力學系統(tǒng)最重要的性質(zhì)之一，在數(shù)學和工程領域得到廣泛的研究[1-3].眾所周知，Lyapunov理論是分析穩(wěn)定性最有效和最流行的工具，其核心是構(gòu)造合適的Lyapunov函數(shù)(無時滯系統(tǒng))[4-5]或Lyapunov泛函(時滯系統(tǒng))[6-8].

由于計算技術不斷發(fā)展，二次Lyapunov函數(shù)得到廣泛應用，所得的穩(wěn)定性判據(jù)一般用線性矩陣不等式來描述[9].對時滯系統(tǒng)，二次Lyapunov泛函是分析穩(wěn)定性的有力工具[10-13].然而在很多情況下，用二次Lyapunov函數(shù)或二次Lyapunov泛函獲得低保守性的穩(wěn)定性判據(jù)相當不易[14-15].因此，需要尋求穩(wěn)定性分析的新方法.最近，協(xié)正多項式Lyapunov函數(shù)(齊次多項式Lyapunov函數(shù)的一種特殊形式)被用于任意切換信號的切換系統(tǒng)[16].Chesi等人提出一種無保守性線性矩陣不等式條件驗證滿足滯留時間的切換系統(tǒng)的指數(shù)穩(wěn)定性[17]，這啟發(fā)了廣大學者構(gòu)造高次多項式Lyapunov函數(shù)，而不是二次Lyapunov函數(shù)，分析動力學系統(tǒng)的穩(wěn)定性[17-20].

在此背景下，人們開始用多項式Lyapunov泛函研究時滯系統(tǒng).然而，多項式Lyapunov泛函方法尚未得到深入研究.文獻[21-22]嘗試采用該方法建立時滯系統(tǒng)的穩(wěn)定性條件.需要注意的是，這兩篇文獻的主要推證有誤.因此，本文將進一步探索多項式Lyapunov泛函.

本文結(jié)構(gòu)安排如下:第1節(jié)介紹了預備知識，第2節(jié)給出主要結(jié)果，第3節(jié)給出一個數(shù)值例子，第4節(jié)總結(jié)全文.

1 問題陳述及預備知識

2 主要結(jié)果

3 數(shù)值例子

本節(jié)給出一個數(shù)值例子驗證所得的理論結(jié)果.

考慮下面的系統(tǒng)方程:

表1 時滯的上界:時變時滯(q＝2)Table 1 Upper bound of delays:Time-varying delays(q＝2)

4 結(jié)論

本文針對時滯系統(tǒng)提出一種齊次多項式Lyapunov泛函方法，建立了系統(tǒng)的穩(wěn)定性條件.數(shù)值例子表明本文給出的方法對快速變化的時滯顯著效果.

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